The one-phase Hele–Shaw model that we are concerned with has been applied to three main configurations, namely an expanding bubble of air into an infinite body of fluid, a contracting finite blob of fluid, and the displacement of viscous fluid in a Hele–Shaw channel. In each of these three scenarios, the fluid boundary is unstable (the Saffman–Taylor instability), and a typical outcome involves portions of the interface propagating increasingly faster than other portions and in some cases leading to a striking fingering pattern at the boundary. For the special zero-surface tension case (also known as Laplacian growth), a host of mathematical studies based mostly on conformal mapping, conserved moments and the Baiocchi transform have highlighted the possible scenarios for this ill-posed model, including exact solutions and finite-time blow-up [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}.
For the more physically realistic nonzero surface tension case (which is well-posed), the broader strategies to study this problem include stability analysis [13]}, [14]}, small-surface-tension asymptotics [15]}, [16]}, employing harmonic moments and conserved quantities [17]} and fully numerical methods mostly with boundary integral methods [18]}, [19]}, [20]}, [21]}, [22]}, [23]} but also the level set formulation [24]}.
While we shall devote much of our attention in this article to certain non-standard variations of (REF )-(REF ), we do not provide any commentary on how the boundary conditions (REF ) may be altered by considering additional physical effects on the boundary apart from surface tension, including the effect of a dynamic contact angle and the related issue of kinetic undercooling [25]}, [26]}, [27]}, [28]}, [29]}, [30]}, [31]}, [32]}, [33]}, [34]}.
Similarly, we do not review non-Newtonian flows, which themselves are well-studied [35]}, [36]}, [37]}, [38]}, [39]}, [40]}.
Finally, our focus is on time-dependent problems and so we are not intending to review the extensive literature on travelling wave problems involving a steadily propagating finger [41]}, [42]}, [43]}, [44]}, [45]}, [46]}, [33]}, [48]}, [49]}, [50]} or bubble [51]}, [52]}, [53]}, [54]}, [55]}, [56]}, [57]}, [58]}.
